Seasonal effects on the stoichiometry of microbes, primary production, and nutrient cycling


We develop a compartment model inspired by producer–herbivore–microbe soil food webs and determine how the naturally occurring seasonal variation in producer and detrital quality affects microbial nutrient cycling and the feedback to primary production. We show that seasonal changes in the stoichiometric quality of the producer coupled with the efficiency of herbivore grazing could induce a switch in the stoichiometric signature and therefore the functioning of the microbial community. Microbial decomposers are responsible for the flux of essential nutrients through an ecosystem. Our model enables one to quantitatively understand the tipping points between bacterially or fungally dominated decomposer communities, and more generally, the complex relationships between microbial decomposers, primary production, and nutrient cycling.

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KC was supported by National Science Foundation award CMMI-1233397 and by a Margaret and Herman Sokol Research Award. EF and LB were supported by National Science Foundation award DMS-1853610. JAK was supported by National Science Foundation award CBET-1603741. We warmly thank Leah B. Shaw for reading and commenting on an early version of the manuscript.

Author information




KC, EF, and JAK developed the model. KC, EF, and LB performed the analysis. All authors participated in writing the manuscript.

Corresponding author

Correspondence to Eric Forgoston.

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The authors declare that they have no conflict of interest.

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All authors gave final approval for publication.


Appendix A: Detritus C/N ratio, \(\mu\)

The detritus C/N ratio, \(\mu\), is a function of the producer C/N ratio \(\alpha\) and the herbivore C/N ratio \(\gamma\). Since the detritus C/N ratio \(\mu = C_{D}/N_{D}\), one has

$$\begin{aligned} \mu \frac{dN_{D}}{dt} = \frac{dC_{D}}{dt}. \end{aligned}$$

Substitution of Eqs. (18)-(19) leads to

$$\begin{aligned}&\mu \Bigl [ d_P N_P + d_H N_H + (1-e)hN_HN_P - l_DN_D \\&\quad - Min \left( m_NN_D, \dfrac{\delta }{\mu - \delta }r_IN_I \right) \Bigr ]\\= & \ \alpha d_PN_P + \gamma d_HN_H + \alpha (1-e)hN_HN_P \\&- \mu l_DN_D - \mu \cdot Min \left( m_NN_D, \dfrac{\delta }{\mu - \delta }r_IN_I \right) . \end{aligned}$$

Simplification leads to

$$\begin{aligned}&\quad \mu \left[ d_PN_P + d_HN_H + (1-e)hN_HN_P \right] \\&= \alpha d_PN_P + \gamma d_HN_H + \alpha (1-e)hN_HN_P, \end{aligned}$$

and solving for \(\mu\) one finds that

$$\begin{aligned} \mu = \dfrac{\alpha d_{P} N_{P} + \gamma d_{H} N_{H} + \alpha (1 - e) h N_{H} N_{P}}{d_{P} N_{P} + d_{H} N_{H} + (1 - e) h N_{H} N_{P}}. \end{aligned}$$

We can rearrange the expression for \(\mu\) in terms of \(\alpha\) and \(\gamma\) to represent the detritus C/N ratio as a weighted average of the producer and herbivore mass as follows

$$\begin{aligned} \mu= & \ \alpha \left( \dfrac{d_{p} N_{P} + (1 - e) h N_{H} N_{P}}{d_{p} N_{P} + d_{H} N_{H} + (1 - e) h N_{H} N_{P}}\right) \\&+ \gamma \left( \dfrac{d_{H} N_{H}}{d_{p} N_{P} + d_{H} N_{H} + (1 - e) h N_{H} N_{P}} \right) . \end{aligned}$$

The coefficient \(\alpha\) represents the percentage of detritus coming from the producers, while the coefficient \(\gamma\) represents the percentage coming from the herbivores.

Appendix B: C-Limited Decomposer Detritus Uptake Rate, \(m_{N}\)

The uptake rate \(m_{N}\) is based on the equation for \(\mu\) so that

$$\begin{aligned} m_{N}= & \ a \left( \dfrac{d_{p} N_{P} + (1 - e) h N_{H} N_{P}}{d_{p} N_{P} + d_{H} N_{H} + (1 - e) h N_{H} N_{P}} \right) \\&+ j \left( \dfrac{d_{H} N_{H}}{d_{p} N_{P} + d_{H} N_{H} + (1 - e) h N_{H} N_{P}}\right) , \end{aligned}$$

where a is the C-limited decomposer uptake rate of plant detritus, and j is the C-limited decomposer uptake rate of herbivore detritus. Because the model is already so complex, a fixed value of \(m_{N}\) was used (Cherif and Loreau 2013).

Appendix C: Steady States, Decomposer Biomass, and Initial Conditions

Analytical steady states and the corresponding Jacobian matrix can be found for the original five equation model using Mathematica. However, the analytical steady states and Jacobian are extremely long and complicated and take up many pages of space. Since they are not illuminating, they are therefore not included in the article or appendices.

To numerically solve the system of equations, we must determine the decomposer biomass and initial conditions for the system. First, numerical steady states were found for the original five equation model in a C-limited state. A carbon limited state was chosen because while C-limited, it is possible for decomposers to mineralize nitrogen to the inorganic compartment or immobilize needed nitrogen from the inorganic compartment. In contrast, N-limited decomposers can only immobilize nitrogen, which could unnecessarily limit our system. We used the midpoint values for plant C/N, herbivore efficiency, and decomposer C/N, (\(\alpha = 30\), \(e = 0.55\), and \(\beta = 7\)), and herbivore respiration rate \(r_H = 0.014\) (Krumins 2015). Herbivore C/N ratios can reasonably vary between 7 and 10 in a forest setting (Krumins 2015; Cherif and Loreau 2013), so a midpoint herbivore C/N ratio of \(\gamma = 8.5\) was used. The numerical steady state for the decomposer N compartment gives \(N_M = 7.480\).

For this value of \(N_M\) and our starting conditions on June 21 (\(\alpha = 20\) and \(e = 0.8\)), the steady states of the mass balance equations with the exception of the decomposer equation were found numerically. These numerical steady states, shown in Table 4, are used as the initial values for the system. The system was solved numerically using a fourth-order Runge–Kutta solver in MATLAB.

In the numerical solver, the system was allowed to switch between C and N limitation using Liebig’s Law of the Minimum, employing a small threshold of 0.00005. Because possible decomposition values are in many cases extremely close in value, a switch between C and N limitation could occur at every time step, resulting in several switches per day, which is biologically unrealistic. Further, a switch between limitations causes discontinuities in decomposition and decomposer demand values due to the change in their definitions. The thresholding provides a realistic mechanism to determining when a switch occurs.

Table 4 Numerically computed steady-state values used as initial conditions

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Carfora, K., Forgoston, E., Billings, L. et al. Seasonal effects on the stoichiometry of microbes, primary production, and nutrient cycling. Theor Ecol (2021).

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  • Plant-herbivore-microbe interactions
  • Seasonal effects
  • Bacterial–fungal cycling
  • Stoichiometry